Exercise 16, § 8

(a) Let A be a closed countable subset of R, f a continuous numerical function defined in R. If f is constant on each of the intervals contiguous to A, prove that f is constant (use the Bolzano theorem).

(b) If f is the continuous mapping of the Cantor ternary set K into [0, 1], defined in Exercise 9 (IV, p. 63), prove that one can extend f to a continuous function in R, and constant on each of the intervals contiguous to K.